proof of extended Liouville’s theorem

This is a proof of the second, more general, form of Liouville’s theorem given in the parent (http://planetmath.org/LiouvillesTheorem2) article.

Let $f:\mathbb{C}\to\mathbb{C}$ be a holomorphic function such that

$|f(z)|<c\cdot|z|^{n}$

for some $c\in\mathbb{R}$ and for $z\in\mathbb{C}$ with $|z|$ sufficiently large. Consider

$g(z)=\begin{cases}\frac{f(z)-f(0)}{z}&z\neq 0\\ f^{\prime}(0)&z=0\end{cases}$

Since $f$ is holomorphic, $g$ is as well, and by the bound on $f$ , we have

$|g(z)|<c_{1}+c_{2}\cdot|z|^{n-1}<c^{\prime}\cdot|z|^{n-1}$

again for $|z|$ sufficiently large.

By induction, $g$ is a polynomial of degree at most $n-1$ , and thus $f$ is a polynomial of degree at most $n$ .

Title	proof of extended Liouville’s theorem
Canonical name	ProofOfExtendedLiouvillesTheorem
Date of creation	2013-03-22 16:18:31
Last modified on	2013-03-22 16:18:31
Owner	rm50 (10146)
Last modified by	rm50 (10146)
Numerical id	8
Author	rm50 (10146)
Entry type	Proof
Classification	msc 30D20